1. 2.6 Rational Functions and Asymptotes 2.7 Graphs of Rational Functions Rational function – a fraction where the numerator and denominator are polynomials

2. Domain The denominator can never equal zero all realsDomain includes all reals except where the denominator is zero Vertical Asymptotes occur where the denominator is zero Asymptotes are lines that the curve approaches but does not cross

3. Horizontal Asymptotes The graph of f(x) has at most one horizontal asymptote determined by comparing the degrees of the numerator (n) and denominator (m) If n<m, the line y=0(x-axis) is a horizontal asymptote If n=m the asymptote is the horizontal line formed by the ratio of the coefficients of the leading terms If n>m the graph of f(x) has no horizontal asymptotes

4. Example: Find the domain and the asymptotes

5. Non-rational graphs Functions that are not rational can have 2 horizontal asymptotes The left end of the graph can approach a different value than the right end of the graph Consider

6. Guidelines for graphing Set the denominator equal to zero and find the domain and vertical asymptotes Find the horizontal asymptotes using the leading terms Sketch the asymptotes as dotted lines on your graph Type the equation into your calculator (watch your parenthesis) and plot points from the table on either side of the asymptotes

7. Graph each Rational function

8. Slant Asymptotes Occur when the degree of the numerator is exactly one more than the degree of the denominator To find the equation of a slant asymptote, divide using long division The slant asymptote equation will be linear (y=mx+b) in form

9. Example: Graph